Wake up and smell the quantum

I had a lot of fun working with Brady Haran on this Sixty Symbols video, uploaded yesterday:

We physicists spend a lot of time talking up the weird and wacky aspects of quantum mechanics — entanglement, teleportation, many worlds, tunnelling, the philosophical ramifications of the wavefunction…, you know the drill. For a change, I wanted to make a video with Brady that highlighted just how many aspects of the quantum world can be explained in terms of phenomena and patterns we’re used to seeing in the world around us; in other words, to ground quantum principles in everyday physics. And what could be more commonplace — some might even say mundane (though not me) — than a cup of coffee?

The video describes the staggering quantum corral images which were created by Mike Crommie, Chris Lutz, and Don Eigler back in the early nineties, as discussed in this ground-breaking paper. (Unfortunately, due to some crossed wires between Brady and me, and largely because I swamped his e-mail inbox with different links to various descriptions of the quantum corral work, the video mistakenly credits Joe Stroscio and Don Eigler — rather than Crommie, Lutz, and Eigler — for the image below. Joe Stroscio has done some phenomenal scanning probe work in his time, but he’s not responsible for the corral.)

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The corral is formed of 48 iron atoms which have been painstakingly put in place, one at a time, using the tip of a scanning tunnelling microscope. (Coincidentally, Joe Stroscio and colleagues have introduced autonomous atom manipulation which allows these types of atomic arrangements to be “dialled in” and fabricated directly under computer control). The ripples that can be seen both inside and outside the corral are due to the variation in electron density across the surface — electron waves scatter off (i.e. are reflected from) the Fe atoms, interfere, and we’re left with a standing wave inside the corral. Because the corral is circular, that standing wave is described mathematically by something known as a Bessel function. And that precise mathematical function also describes the standing wave that forms in a cup of coffee.  Even though the diameter of the coffee cup is roughly six million times larger than that of the corral.

Physicists, and scientists in general, are very used to seeing mathematics describe very many aspects of our reality. This degree of familiarity with the ubiquity of mathematics in nature can sometimes make us — well, at least sometimes makes me — rather too blasé about just how utterly remarkable it is that precisely the same mathematical function can describe behaviour in completely different materials, spanning a huge range of length scales, and in entirely different environments. The only thing that’s common between the cup of coffee and the quantum corral is the symmetry. And yet the coffee and the electrons produce exactly the same pattern. (Well, as long as the critical “sloshing” point for the coffee isn’t reached. There was a great paper in Physical Review E back in 2012 on this topic).

What I don’t say in the video, however, is that there’s something very special about the copper sample on which the Fe atoms are sitting. It’s called a Cu(111) surface, where the numbers, known as Miller indices, describe the direction in which a copper crystal has to be cut to expose that particular plane. (Symmetry is all-important here too). At the Cu(111) surface the electrons are free to move across the plane; we call the system a 2D electron gas (although, in the video, I use the term “electron fluid” to bring out the comparison with the coffee. This isn’t such a “reach” – the term Fermi liquid is used throughout solid state physics). Not all surfaces give electrons this freedom to roam. The corral experiment would never work on a Si(111) surface, for example, because the electrons there, due to the strong covalent bonding in the crystal, simply don’t have the same leeway to explore the space around them.

I’ve written before that I’ve always been impressed that the comments under Sixty Symbols videos buck the usual trend for below the line online commentary, particularly at YouTube: the points the Sixty Symbols audience raise are very often insightful, smart, and even erudite at times. This is again true for the “quantum coffee” video. The following comment asks a particularly perceptive question related to what is causing the waves — are the electrons “driven” by the STM tip in some way?

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The current from the STM tip is not responsible for “driving” the pattern. Or, to put it another way, the standing wave state of the electrons is not produced by the probe. Although STM can certainly be used in a very invasive way — this is precisely how the atoms are arranged to form the corral in the first place — it can also be used as relatively non-invasive probe of the electron density. Indeed, the same type of scattering is seen at naturally occuring defects (e.g. atomic step edges), as clearly seen in the image below (also taken from the IBM gallery). The ripples at the step edges are what are called Friedel oscillations and, again, arise from electron waves being back-reflected from the step.

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As is my wont, I sneak a guitar into the video as an example of a one dimensional standing wave, in contrast to the 2D Bessel function pattern. In another key example of the pervasiveness of mathematics, there are particularly striking parallels between waves on a guitar (and other lesser musical instruments) and the quantum world. I’ve banged on about this at length before in the context of the Heisenberg uncertainty principle, so won’t hammer home the point again here. But what you might well ask is whether it’s possible to make a one-dimensional “corral” out of a line of atoms (as opposed to a 2D container).

It is. The image below shows the electron density in a 1D chain of Pd atoms, created and imaged using an STM by Nilius, Wallis and Ho ten years ago and elegantly described in this paper. By applying a different voltage to the STM tip, they can access different electron energies. The patterns of electron density, i.e. the standing waves, that they see as a function of voltage are very similar to those seen for waves on a guitar string. If you want to know more about this, including some of the not-so-gory mathematical detail, I cover it in the 1st year undergraduate Frontiers in Physics module here at Nottingham. Chapter 4 of the ebook for the nanoscience component of the module covers standing waves in the 1D atomic chain.

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Another aspect of 2D standing waves we didn’t explore in the video, but which I’m hoping Brady and I will cover in the not-too-distant future, is the relationship of the quantum corral to drums and drumming. One of my all-time favourite scientific papers had that precise topic as its theme. But I’ll bang that particular drum in a future blog post.

Left in the lurch? On Corbyn, comedy and credibility

This arrived in the post at the beginning of July:

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Yep, I signed up to vote for Jeremy Corbyn in the upcoming Labour leadership elections. (Here’s how to join, if you’re interested. It’s a quick and entirely painless process.) If you’re a UK resident, and unless you’ve been living in an alternate reality for the past month — in, for example, a parallel universe whose inhabitants still give a toss about Tony Blair’s proclamations — you’ll know that Corbyn has had a meteoric rise to the top of the Labour leaders’ board. Yesterday he was named as the bookies’ favourite, at 5-4 odds; six weeks ago he was a 100-1 outsider. This is the “biggest price fall in political betting history” according to William Hill.

This eloquent and compelling piece lays out many of the reasons why I’m voting for Corbyn. (It doesn’t, however, mention that he’s a staunch republican, having petitioned Blair to remove the Royal family from Buckingham Palace and place them in “more modest” accomodation. For that alone he’d get my vote. (And, yes, before you ask, I know about the homeopathy thing. Bear with me, I’ll get to it in a future post.)).

But according to a slew of articles in The Guardian and The Observer over the last few weeks, I’m a narcissistic, deluded, reactionary, dogmatic, immature, head-in-sand (and foot-in-sandal), tribal, ideologically-driven, confused lefty dinosaur for even beginning to entertain the slightest inkling of an idea that Corbyn’s leadership challenge might possibly be a good thing for not only the Labour party but for the entire country.

I’ve got used to reading those articles over my bowl of muesli in the mornings but Wednesday’s Guardian upped the ante just that little bit too far. In a piece claiming that Corbyn was humourless for having the temerity to say that, should he win, he’d like Lennon’s Imagine played at his victory rally, Jason Sinclair — yeah, me neither — made the truly remarkable claim that “We demand our politicians can display their common sense by telling good jokes“.

Errmm, what?

No, really. What?

It turns out that Sinclair, a copywriter, is responsible for the @corbynjokes Twitter feed, the focus of the article he wrote for The Guardian. To be fair to Sinclair, his feed generated one OK joke. This one:

Sinclair must have been spending quite some time in his own peculiar parallel universe, however, if he thinks that politicians tell good jokes. Either that or his threshold for what he considers good comedy is startlingly low. (Perhaps he moonlights as a Radio 4 sitcom writer?)

I’m going with the latter explanation. Here’s why. Another line from Sinclair’s article…

“Boris Johnson is a major political force in part because he has passable comic delivery.”

Hmmm. No politician, including Johnson, has ever made me laugh as a result of their comic delivery. And I’m not talking about gut-busting, tears rolling down cheeks, rolling on the floor laughter. Nor a hearty chuckle. Or even a knowing, spontaneous giggle. Indeed, I’d be more than happy if a politician’s joke could coerce even a weak smile from me every now and again. Instead, politicians’ attempts at humour are invariably so arse-clenchingly, toe-curlingly, cringe-makingly, gob-smackingly embarrassing that my natural reaction is to die a little inside on their behalf.

Now, the explanation for my lack of appreciation of, as Sinclair would have it, the natural comedic flair of our political class could be, of course, that I’m a dour, humourless, bearded lefty gobshite who is genetically incapable of cracking a smile occasionally. While I’d freely admit that grumpiness is not exactly a stranger to me, there are quite a few exceptionally talented writers out there whose well-observed, intelligent, witty, and original insights regularly crack me up. One of these is Charlie Brooker, who has written a wonderfully acerbic weekly column for the Guardian for many years. Here’s what Brooker had to say about a certain tousle-haired toss..  politician back in 2008:

On May 1 London chooses its mayor, and I’ve got a horrible feeling it might pick Boris Johnson for similar reasons. Johnson – or to give him his full name, Boris LOL!!!! what a legernd!! Johnson!!! – is a TV character loved by millions for his cheeky, bumbling persona. Unlike the cartoon MP, he’s magnetically prone to scandal, but this somehow only makes him more adorable each time. Tee hee! Boris has had an affair! Arf! Now he’s offended the whole of Liverpool! Crumbs! He used the word “picaninnies”! Yuk yuk! He’s been caught on tape agreeing to give the address of a reporter to a friend who wants him beaten up! Ho ho! Look at his funny blond hair! HA HA BORIS LOL!!!! WHAT A LEGERND!!!!!!

Copywriters are not exactly renowned for their originality. Like many of their colleagues in marketing and advertising, they have a reputation for churning out retreads of bland boilerplate with little or no creative copy. (See Private Eye, passim). This might help explain why Sinclair’s expectations when it comes to insightful and intelligent comedy are so low – he’s working in a field where wit is the exception rather than the norm.

Marketing, advertising, and copywriting are too often the living dead embodiment of style over substance, responsible for the type of banal bollocks designed to appeal to those who are entirely at ease with cliched, vacuous tripe, i.e. New Labour’s (and the Blairites’) stock-in-trade.

I prefer some substance to my politics. And to my comedy.

Here’s Bill Hicks on the subject of marketing.

Pushing the potential of probe microscopy

Christian Wagner and co-workers at Forschungszentrum Jülich have developed an elegant and exciting way of mapping the electrostatic potential of single atoms and molecules with very high sensitivity. Their paper on this new approach, which they call scanning quantum dot microscopy, was published in Physical Review Letters yesterday. I was asked by the American Physical Society to write a Viewpoint article on Wagner et al.’s paper and jumped at the invitation because I found the work so fascinating and important. Here’s my take on what they’ve done: Pushing The Potential of Probe Microscopy

If it looks like a duck…

Last week I attended, and spoke at, a session entitled “Frontiers of Scanning Probe Microscopy” at this year’s Microscience Microscopy Congress in Manchester. The focus of the presentation I gave there — and it’s a recurrent theme in the talks and seminars I give at the moment — was the thorny problem of identifying and interpreting artefacts in images of atoms and molecules.

Microscopists tend to be skeptical about that old maxim, “seeing is believing”. But, as I’ll show below, sometimes we’re simply not skeptical enough. This is not just an issue for those involved in imaging and microscopy — it’s at the core of all science: how do we know our measurements are an accurate picture of reality? (Whichever version of reality we prefer…).

Every image out there, regardless of how it was created, is a convolution of the properties of the object and the characteristics of the imaging system. (And that includes our eyes). The word convolution has its roots in the Latin convolvere, meaning “to roll together”. That’s a great description of the mathematical physics underpinning the process: the functions describing the object and the imaging system are indeed rolled together (via a convolution integral).

Twenty-five years ago, the Hubble telescope gave us a spectacularly (un)illuminating insight into the essence of convolution. The images below, taken from the Wiki page for the HST, vividly show the effects of the convolution process when the imaging characteristics of the telescope were, let’s say, rather poor (on the left) and when they were much improved by the addition of corrective optics (on the right).

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The imaging system — and this holds true for any imaging system, be it a microscope, telescope, camera, or whatever arbitrary combination of optics we put together — is characterised via a very simple concept: the point spread function. That function does exactly what it says on the tin: it captures how the image of a single point in the object spreads in space as a result of the imaging system. We then take the point spread function and apply it in turn to all of the points in an object in order to determine what the resulting image will be. For the HST images above, the point spread function is substantially broader for the image on the left than for that on the right.

I should stress that these types of convolution effect are, of course, not limited to images — they hold for any measurement and any type of signal. Ten years ago, I taught an undergraduate module on Fourier analysis and spent quite some time on convolution. (I’ll save the elegance of the Fourier treatment of convolution for a future post). I used the various sound samples below to show the students how convolution works for an audio, rather than a visual, signal. In this case, the point spread function is the response of the surroundings (be it a cave, lecture theatre, auditorium, forest, classroom…) to a very short, sharp signal: the audio equivalent of a single pixel or point. Think of it like making one short hand-clap in a room: the point spread function, which for audio signals is called the impulse response function, is the sound of that clap reverberating. (Yes, the hand-clap is just an approximation to the type of short, sharp signal — i.e. impulse — we need but it serves to make the point.)

So, let’s take a large concert hall. Here’s the impulse response for the hall:

Now consider a space which is rather less grand (at least in terms of its audio characteristics). An ice cavern, say…

Note the very audible differences between the impulse response for the concert hall and for the cavern.

Now let’s take an audio signal completely at random. Like this…

If we convolve the Pythons’ Gregorian chant with the impulse response for the concert hall, here’s what we get.

And this is the convolution of the chant with the impulse response for the ice cavern:

Just as with the HST images, the response of the system (the concert hall or the ice cavern in this case) can be worked out from its audio “point spread function” (the impulse response).

For scanning probe microscopy (SPM), however, we’re in a whole new world of pain when it comes to deciphering the contribution of the imaging system to the image we see. Far from being a static distortion as in the HST optics, the scanning probe itself responds dynamically to the object under study. The simple point spread function approach breaks down. And this can lead to some very misleading images indeed…

My first love in research was, and will always be, SPM. I’ve written about the power and pitfalls of the technique in detail before but the concept at the heart of the technique is really very simple indeed. (Its execution rather less so). We take an exceptionally sharp probe — terminated in a single atom or molecule — and move it very close to a surface, an interface, or a single atom or molecule. When i say “very close”, I mean within a few atomic diameters, or, in the highest resolution work, about a single atom’s distance from a surface. At those distances a number of forces and interactions come into play including, in particular, chemical bond formation and, as described in this post for the Institute of Physics’ physicsfocus blog last year, electron-electron repulsion due to the Pauli exclusion principle. By scanning the probe back and forth (using piezoelectric motors) we can measure the variation of those forces within a single molecule and convert that signal to an image.

Leo Gross and co-workers at the IBM research labs at Rüschlikon in Zurich pioneered a new sub-field of scanning probe microscopy when they showed back in 2009 that images of the internal architecture of single molecules could be captured. The agreement between these images and the ball-and-stick models used by chemists (and physicists) to represent molecules is striking, to put it mildly. While the picture of the tip in the figure to the right is an artist’s representation, the image of the molecule directly below is the actual experimental data measured for a single pentacene molecule, the ball-and-stick model for which is shown at the foot of the figure (grey spheres are carbon, white spheres are hydrogen — it’s a molecule so simple even physicists can understand it.)

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Ultrahigh resolution images showing submolecular structure in exquisite detail for a variety of molecules followed (as described in this book chapter). But a number of SPM research groups across the world, including our team at Nottingham, were particularly keen to ascertain whether intermolecular bonds (rather than, or in addition to, intramolecular bonds) could be resolved using the method introduced by IBM Zurich. Nottingham has a long track record — through the efforts of the research groups of my colleagues Peter Beton and Neil Champness — of exploiting hydrogen bonds in the assembly of supramolecular systems. Hydrogen bonds are also of key importance in biochemistry, including, of course, in underpinning base pair interactions in DNA. Could we actually see hydrogen bonds between molecules using probe microscopy?

We started the experiments.

And we were over the moon when we acquired this image of a hydrogen-bonded lattice of molecules shortly afterwards…

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 …particularly as we could apparently map the “filamentary” features between the molecules directly onto where we expected the hydrogen bonds (the dotted lines in the image below) to be:

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While we were puzzling over how to interpret the image — just why did the hydrogen bonds appear so bright compared to the bonds inside the molecules? — we were somewhat less over the moon to be scooped on the first ‘observation’ of hydrogen bonds, as described in the article below. (Click on the image for the full Chemistry World piece).

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Note the social media hits on that article: the images certainly created a stir.

Once more it looks like there’s exceptionally good agreement between the positions of the hydrogen-bond features in the probe microscope image on the left above and those expected on the basis of the chemistry (as sketched in the diagram to the right).

But, again, why are the H-bonds so bright in the image? The authors’ own calculations showed that the electron density between the molecules could not account for the brightness — there just wasn’t enough charge there. (This chimed with our experience, as we described in a paper published a few months later. (Free to read — no paywall)).

Even though the positions of the features in the images met all of our expectations with regard to where we’d expect hydrogen bonds to be observed, was it possible that it was simply some type of image artefact? Were those ‘bonds’ nothing more than nanoscopic will-o’-the-wisps? Could Nature really be that cruel? (That’s Nature as in the universe around us, not Nature the journal. Scientists all know just how cruel Nature can be…).

Yes, Nature is that cruel.

It turns out that the intermolecular features readily appear in simulations based around the type of simple interatomic potentials we explain to our 1st year undergraduate students. (See, for example, the sections on Lennard-Jones and Morse potentials in Chapter 2 of this ebook). The simulations in question know nothing about the electron density due to bonding between the molecules — they are based solely on the atomic coordinates, i.e. the positions of the atoms in the molecules. And yet, as the images below show, the simulations — which have been developed by a number of groups in parallel, particularly those of Pavel Jelinek at the Academy of Sciences of the Czech Republic and Peter Liljeroth at Aalto University in Finland — provide exceptionally good agreement with the experimental images. (The figure below is taken from this paper by Prokop Hapala and co-workers in Jelinek’s group, along with a team at Forschungszentrum Juelich comprising Stefan Tautz, Ruslan Termirov, Christian Wagner and Georgy Kichin).

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So if we’re not really seeing bonds, just what is going on?

When acquiring ultrahigh resolution images of the type pioneered by Leo Gross and colleagues, it is generally the case that the tip is terminated — either deliberately or inadvertently — with a single molecule. (The eagle-eyed among you might have noticed the CO molecule hanging off the end of the tip in the artist’s impression of the pentacene imaging experiment above). The molecular probe can flex and pivot as it is dragged across the target molecule — the apex of the tip bends back and forth due to the forces it experiences from the atoms of the molecules underneath. And that bending motion gives rise to the intermolecular features.

No intermolecular bonds required.

In a clever experimental design, Sampsa Hämäläinen, Liljeroth and co-workers used a molecule which forms hydrogen bonds at some places, but not at others, to highlight the exceptionally important role of the probe in generating spurious intermolecular features. The same type of effect has also been observed for halogen bonding and, most recently, for a system where no intermolecular bonds at all are expected: a lattice of buckyballs (C60 molecules). (I presented these latter data at the conference in Manchester.)

What’s more, and to add to the pain, even if we ‘lock down’ the probe molecule in the simulations — which we did for our calculations — so that it can’t flap around, we’re still left with the point spread function to contend with. The probe has a finite width (in terms of its electron ‘cloud’) and, as pointed out by Hapala and colleagues, this can also generate artefacts via convolution between the probe and the target molecule.

Douglas Adams once said

If it looks like a duck, and quacks like a duck, we have at least to consider the possibility that we have a small aquatic bird of the family anatidae on our hands.

It’s indeed possible. But when it comes to science, it can look like a duck, waddle like a duck, and quack like a duck…

…but all too often it can be a goose.